The Riemannian gradient [1]} at \({\Sigma }\in \mathcal {S}^{++}_p\)  of an objective function \(f:\mathcal {S}^{++}_p\rightarrow \mathbb {R}\)  is the only tangent vector such that for all \({\xi }\in \mathcal {S}_p\) ,
\(\langle \nabla ^{\mathcal {S}^{++}_p}f({\Sigma }),{\xi }\rangle ^{\mathcal {S}^{++}_p}_{{\Sigma }} = f̥({\Sigma })[{\xi }],\) 
